Problem: The graph of $y = ax^2 + bx + c$ has a maximum value of 54, and passes through the points $(-2,0)$ and $(4,0).$  Find $a + b + c.$
Answer: Since the graph passes through the points $(-2,0)$ and $(4,0),$ the equation is of the form $a(x + 2)(x - 4).$

The graph has a maximum, and this maximum value occurs at the average of $-2$ and 4, namely $x = \frac{-2 + 4}{2} = 1.$  But $a + b + c$ is exactly the value of $y = ax^2 + bx + c$ at $x = 1,$ so $a + b + c = \boxed{54}.$